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Page 1 MAGNETIC EFFECT OF CURRENT  III 1. Cyclotron 2. Ampere’s Circuital Law 3. Magnetic Field due to a Straight Solenoid 4. Magnetic Field due to a Toroidal Solenoid Page 2 MAGNETIC EFFECT OF CURRENT  III 1. Cyclotron 2. Ampere’s Circuital Law 3. Magnetic Field due to a Straight Solenoid 4. Magnetic Field due to a Toroidal Solenoid N S D 1 D 2 + W B Cyclotron: D 1 , D 2 – Dees N, S – Magnetic Pole Pieces W – Window B  Magnetic Field H F Oscillator D 2 D 1 Working: Imagining D 1 is positive and D 2 is negative, the + vely charged particle kept at the centre and in the gap between the dees get accelerated towards D 2 . Due to perpendicular magnetic field and according to Fleming’s Left Hand Rule the charge gets deflected and describes semicircular path. When it is about to leave D 2 , D 2 becomes + ve and D 1 becomes – ve. Therefore the particle is again accelerated into D 1 where it continues to describe the semicircular path. The process continues till the charge traverses through the whole space in the dees and finally it comes out with very high speed through the window. W B Page 3 MAGNETIC EFFECT OF CURRENT  III 1. Cyclotron 2. Ampere’s Circuital Law 3. Magnetic Field due to a Straight Solenoid 4. Magnetic Field due to a Toroidal Solenoid N S D 1 D 2 + W B Cyclotron: D 1 , D 2 – Dees N, S – Magnetic Pole Pieces W – Window B  Magnetic Field H F Oscillator D 2 D 1 Working: Imagining D 1 is positive and D 2 is negative, the + vely charged particle kept at the centre and in the gap between the dees get accelerated towards D 2 . Due to perpendicular magnetic field and according to Fleming’s Left Hand Rule the charge gets deflected and describes semicircular path. When it is about to leave D 2 , D 2 becomes + ve and D 1 becomes – ve. Therefore the particle is again accelerated into D 1 where it continues to describe the semicircular path. The process continues till the charge traverses through the whole space in the dees and finally it comes out with very high speed through the window. W B Theory: The magnetic force experienced by the charge provides centripetal force required to describe circular path. mv 2 / r = qvB sin 90° (where m – mass of the charged particle, q – charge, v – velocity on the path of radius – r, B is magnetic field and 90° is the angle b/n v and B) v = B q r m If t is the time taken by the charge to describe the semicircular path inside the dee, then t = p r v or t = p m B q Time taken inside the dee depends only on the magnetic field and m/q ratio and not on the speed of the charge or the radius of the path. If T is the time period of the high frequency oscillator, then for resonance, T = 2 t or T = 2pm B q If f is the frequency of the high frequency oscillator (Cyclotron Frequency), then f = 2pm B q Page 4 MAGNETIC EFFECT OF CURRENT  III 1. Cyclotron 2. Ampere’s Circuital Law 3. Magnetic Field due to a Straight Solenoid 4. Magnetic Field due to a Toroidal Solenoid N S D 1 D 2 + W B Cyclotron: D 1 , D 2 – Dees N, S – Magnetic Pole Pieces W – Window B  Magnetic Field H F Oscillator D 2 D 1 Working: Imagining D 1 is positive and D 2 is negative, the + vely charged particle kept at the centre and in the gap between the dees get accelerated towards D 2 . Due to perpendicular magnetic field and according to Fleming’s Left Hand Rule the charge gets deflected and describes semicircular path. When it is about to leave D 2 , D 2 becomes + ve and D 1 becomes – ve. Therefore the particle is again accelerated into D 1 where it continues to describe the semicircular path. The process continues till the charge traverses through the whole space in the dees and finally it comes out with very high speed through the window. W B Theory: The magnetic force experienced by the charge provides centripetal force required to describe circular path. mv 2 / r = qvB sin 90° (where m – mass of the charged particle, q – charge, v – velocity on the path of radius – r, B is magnetic field and 90° is the angle b/n v and B) v = B q r m If t is the time taken by the charge to describe the semicircular path inside the dee, then t = p r v or t = p m B q Time taken inside the dee depends only on the magnetic field and m/q ratio and not on the speed of the charge or the radius of the path. If T is the time period of the high frequency oscillator, then for resonance, T = 2 t or T = 2pm B q If f is the frequency of the high frequency oscillator (Cyclotron Frequency), then f = 2pm B q Maximum Energy of the Particle: Kinetic Energy of the charged particle is K.E. = ½ m v 2 = ½ m ( B q r m ) 2 = ½ B 2 q 2 r 2 m Maximum Kinetic Energy of the charged particle is when r = R (radius of the D’s). = ½ B 2 q 2 R 2 m K.E. max The expressions for Time period and Cyclotron frequency only when m remains constant. (Other quantities are already constant.) m = m 0 [1 – (v 2 / c 2 )] ½ If frequency is varied in synchronisation with the variation of mass of the charged particle (by maintaining B as constant) to have resonance, then the cyclotron is called synchro – cyclotron. If magnetic field is varied in synchronisation with the variation of mass of the charged particle (by maintaining f as constant) to have resonance, then the cyclotron is called isochronous – cyclotron. NOTE: Cyclotron can not be used for accelerating neutral particles. Electrons can not be accelerated because they gain speed very quickly due to their lighter mass and go out of phase with alternating e.m.f. and get lost within the dees. But m varies with v according to Einstein’s Relativistic Principle as per Page 5 MAGNETIC EFFECT OF CURRENT  III 1. Cyclotron 2. Ampere’s Circuital Law 3. Magnetic Field due to a Straight Solenoid 4. Magnetic Field due to a Toroidal Solenoid N S D 1 D 2 + W B Cyclotron: D 1 , D 2 – Dees N, S – Magnetic Pole Pieces W – Window B  Magnetic Field H F Oscillator D 2 D 1 Working: Imagining D 1 is positive and D 2 is negative, the + vely charged particle kept at the centre and in the gap between the dees get accelerated towards D 2 . Due to perpendicular magnetic field and according to Fleming’s Left Hand Rule the charge gets deflected and describes semicircular path. When it is about to leave D 2 , D 2 becomes + ve and D 1 becomes – ve. Therefore the particle is again accelerated into D 1 where it continues to describe the semicircular path. The process continues till the charge traverses through the whole space in the dees and finally it comes out with very high speed through the window. W B Theory: The magnetic force experienced by the charge provides centripetal force required to describe circular path. mv 2 / r = qvB sin 90° (where m – mass of the charged particle, q – charge, v – velocity on the path of radius – r, B is magnetic field and 90° is the angle b/n v and B) v = B q r m If t is the time taken by the charge to describe the semicircular path inside the dee, then t = p r v or t = p m B q Time taken inside the dee depends only on the magnetic field and m/q ratio and not on the speed of the charge or the radius of the path. If T is the time period of the high frequency oscillator, then for resonance, T = 2 t or T = 2pm B q If f is the frequency of the high frequency oscillator (Cyclotron Frequency), then f = 2pm B q Maximum Energy of the Particle: Kinetic Energy of the charged particle is K.E. = ½ m v 2 = ½ m ( B q r m ) 2 = ½ B 2 q 2 r 2 m Maximum Kinetic Energy of the charged particle is when r = R (radius of the D’s). = ½ B 2 q 2 R 2 m K.E. max The expressions for Time period and Cyclotron frequency only when m remains constant. (Other quantities are already constant.) m = m 0 [1 – (v 2 / c 2 )] ½ If frequency is varied in synchronisation with the variation of mass of the charged particle (by maintaining B as constant) to have resonance, then the cyclotron is called synchro – cyclotron. If magnetic field is varied in synchronisation with the variation of mass of the charged particle (by maintaining f as constant) to have resonance, then the cyclotron is called isochronous – cyclotron. NOTE: Cyclotron can not be used for accelerating neutral particles. Electrons can not be accelerated because they gain speed very quickly due to their lighter mass and go out of phase with alternating e.m.f. and get lost within the dees. But m varies with v according to Einstein’s Relativistic Principle as per Ampere’s Circuital Law: The line integral B . dl for a closed curve is equal to µ 0 times the net current I threading through the area bounded by the curve. ? ? B . dl = µ 0 I ? B . dl = ? B . dl cos 0° ? B . dl = B = ? dl = B (2p r) = ( µ 0 I / 2p r) x 2p r ? B . dl = µ 0 I I B B r O dl I Current is emerging out and the magnetic field is anticlockwise. Proof:Read More
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